𝟐. 𝔊𝔢𝔫𝔢𝔯𝔞𝔩 ℌ𝔦𝔰𝔱𝔬𝔯𝔦𝔬𝔤𝔯𝔞𝔭𝔥𝔦𝔠𝔞𝔩 ℜ𝔢𝔪𝔞𝔯𝔨𝔰

𝔐𝔞𝔫𝔶 of the recent historical words dealing with the emergence of set theory are of a biographical character, Certainly the biographical approach to history has to its strength, but it also was weaknesses. It becomes quite difficult to avoid a certain partially as an effect of excessive concentration on a single author, or simply due to empathic identification.¹ From what precedes, it should be clear that the present winter has made an option for a less narrowly focused historical treatment ― a collective approach. It is almost self-evident that great scientific contributions are collective work, and the emergence of set theory and the set-theoretical approach to mathematics is no exception. 

ℑ𝔫 Part One and Two, the work of a small group of authors is studied through a 'micro' approach, and the peculiarities of their orientation are analyzed by comparison with competing schools or approaches. As much attention is paid to informal communications (personal contacts, correspondence) as publications. Questions of influence thus become central, although I admit that a study of such questions must enter into the domain of the hypothetical much more than a more classical study of published work. Hopefully the reader will acknowledge the value of an inquiry of this kind. 

ℑ have already mentioned that the reader will notice a gradual narrowing of the conceptual issues treated as the narrative develops from the emergence of the set-theoretical approach (Part One) to the foundations of abstract set theory (Part Three). This step-by-step concentration goes in counterpoint with a progressive widening of the group of actors. Set theory began as the brainchild of just a few original thinkers, and it gradually became a community enterprise; as a natural side-effect, the amount of work that the historian has to cover increases exponentially. 

𝔗𝔥𝔢 early history of set theory can only be adequately written when one abandons present disciplinary boundaries and makes at least some attempts to consider cross-disciplinary interactions. One has to pay attention to ideas and results proposed in several different branches of mathematics, if only because mathematics of the late 19th century were by no means as narrowly specialized as they tend to be today.² I must be taken into account that clear boundaries between various branches of mathematics were not yet institutenalized before 1900. The careers of the figures that we shall study are clear examples that specialization was only beginning, and thag 19th century mathematicians enjoyed a great freedom to move from branch to branch. 

𝔅𝔲𝔱 even granting these premises, there are still different ways to approach the history of mathematics. As I see it, my own approach tends to concentrate on the development of mathematical knowledge — the process of invention-discovery, the evolution of views held by mathematicians (both single individual and communities), the research programs that the historical actors tried to advance, the schools and traditions that influenced their work. In this connection, it is convenient to clarify a few general historiographical notions that will be used in the sequel. 

Already in the 19th-century it was common to speak of scientific and mathematical 'schools', although the meaning of the term differed somewhat from present usage in the history of science. Frequently the term carried a pejorative connotation, suggesting a one-sided orientation with excessive attention to some speciality, as happened when some authors referred to the Berlin school. Here we shall employ the word school exclusively in the customary sense of recent historiography, which started over two decades ago with a well-known paper of J.B Morrell.³ In the present context, a research school is a group led normally by only one mathematician, localized within a single institutional setting, and counting on a significant supply of advanced students. As a result of continuous social interaction and intellectual collaboration, members of a school come to share conceptual viewpoints and research orientations. Philosophical or methodological ideas concerning how to do their research, heuristic views regarding what problems are worth being pursued, which paths are dead-ends, and so on. Research schools are not governed by written regulations, they emerge spontaneously as an implicit reciprocal agreement between professor and students "to form a symbiotic learning and working environment based on the research interests of the professor" [Rowe 2002]. Schools are natural units with in larger institutions, such as universities and faculties. Their great importance comes from the fact that they seem to constitute the crucial link between the social and the cognitive vectors of mathematical (or scientific) work and research. 

𝔒𝔫𝔢 can mention several prominent examples of mathematical school in 19th century Germany, like Jacobi's Königsberg school, the school of Clebsch, and Klein's Leipzig school but the most famous one is the so called Berlin school. Some characteristics of this famous school will be studied in chapter 𝟏 and contrasted with the views held by a group of mathematicians associated with mid century Göttingen. Although I shall keep the traditional denomination 'Berlin school,' at this point I would like to mention a related issue that has been raised recently. David Rowe has suggested that one should differentiate schools from centers, linking the first exclusively with the name of their (single) leaders. According to this, we should speak of the Weierstrass school, not the Berlin school.⁴ Certainly, in the 1880s Weierstrass and Kronecker entertained deep differences, so one should distinguish two schools at that time. But it is still open whether the kind of collaboration established by the Berlin mathematicians (above all Kummer and Weierstrass) in the 1860s and 1870s warrants talks of a single school. Here, I must leave the problem open. 

Ernst Kummer 1810 ― 1893

In chapter 𝐼 we shall speak of a Göttingen 'group' formed by Dirichlet, Riemann and Dedekind. which is not called a school because, though in many respects it had similar characteristics, it lacked a noteworthy output of researches (probably because its temporal duration was very short). In other cases, I shall rename as a tradition what 19th-century authors called a 'school,' for instance the 'combinatorial tradition. '⁵ Talk of a tradition implies that one can find a common research orientation in different actors that do not share a common institutional site, but are linked by traceable influences on each other. One should find a significant amount of shared conceptual elements that may have to do with preference for some basic mathematical notions, judgements concerning significant problems to be studied, methodological views affecting the approach to mathematics, and the like.⁶ 

Karl Weierstrass 1815 ― 1897

𝔒𝔫 the other hand, as I have used the term above, a 'research program' concerns a single individual's research projects, expectations, and preferences. Thus I depart from the meaning given to the term by Lakatos [1970], which seems harmless for at least two reasons. Lakatos's term have never been frequently used, in his sense, by historians of science, and the term is employed here only rarely. Which is not to say that I have not tried to analyze carefully the research programs (in the specified sense) of men like Riemann, Dedekind, Cantor, Zermelo, Russel and so on. Of course, the projects of an individual are deeply influenced by the mathematical situation in which he or she has become a mathematician, that is, by affiliation with a school or the influence of traditions. To give an example, I shall try to show that Cantor deviated from the orientation of the Berlin school due, in good measure, to the influence of the Göttingen tradition, i,e., the orientations emboided in the writings of members of the Göttingen group. 

𝔗𝔥𝔢𝔯𝔢 are other histrographical terms that can be found more or less frequently, but which I shall not employ. It may be useful, hoever, to mention a couple of them in order to clarify the notions above. 'Invisible colleges' are groups of members of a single community or discipline, joined together by formal links (e.g., co-citation) and informal communication. One should reflect that normally we find members of different, and even opposing, research schools in a single invisible college (e.g., Riemann and Weierstrass or Dedekind and Kronecker). It has been suggested that one might focus on a specific kind of invisible college, the 'correspondence network,' that would be particularly important in the case of mathematics [Kushner 1993]. Most of the actor studied here were, in fact, linked to each other through several correspondence networks, and it may be useful to read the following pages with that in mind. But the notion in question will not be explicitly used. 

ℑ stated above that my approach stresses the development of mathematical knowledge ― how it was obtained, refined, and generally adopted. It is my hope that, by paying attention to these issues, it will become possible to unearth the multiple connections between set theory and the boarder context of modern mathematics. Set theory emerged as part of an evolving new picture of the discipline, incorporating a more conceptual and decidedly abstract approach to traditional problems. It has been one of my goals to afford an understanding of the novelties of that approach and the opposition and difficulties it had to face. In this way i also hope to contribute to a richer understanding of the 'classical' world of 19th century mathematics. Standard historiography, with its tendency to project present day conceptions (and myths), has frequently acted as a barrier cutting off a satisfactory interpretation both of the past and a road to our present. 

𝔄𝔫𝔬𝔱𝔥𝔢𝔯 key goal has been to delineate the conceptual shifts affecting disciplines and notions that are frequently taken for granted ― e.g., the very notion of logic and the relation between logic and set theory. Such conceptual changes become a fascinating topic and simulate reflection on the basic ideas that we are familiar with, calling attention to alternative possibilities for theoretical development. I belive the work may turn out interesting for the working mathematician in this particular way. 

ℑ𝔫 writing the present book I have worked as a historian of mathematics, but I would like to mention that, by training, my background is in logic and history & philosophy of science. That may still be visible in the particular orientation I have given to the selection of material, and in the approach that I have preferred to take. For this reason the volume should be of interest to philosophers of mathematics, who will find detailed case studies that offer much valuable material for philosophical reflection. 

𝔗𝔥𝔢 initial impulse came through Javier Ordóñez (Madrid), who introduced me to the history of science and directed my doctoral work; above everything else, I wish to thank this constant support and friendship. During a fruitful stay at the University of California at Berkeley, and later, I've had the opportunity to discuss diversive aspects of the history of set theory with Gregory H. Moore (Hamilton, Ontario), for many years I have also profited from interaction with a leading expert of Dedekind, Ralf Haubrich (Göttingen); he helped me in the most diverse ways during the production of this book. David E.Rowe (Mainz) has been kind enough to revise the present edition and give his expert advice on many issues, big and small. Even though he and others have helped me correct the English, I fear the final version will still show too clearly that the author is not a native speaker. I can only ask the reader for indulgence. 

𝔖𝔢𝔳𝔢𝔯𝔞𝔩 other people have been helpful at different stages in the preparation of this volume: Leo Alonso (Santiago de Compostela), Leo Corry (Tel-Aviv), John W. Dawson, Jr and Cheryl A. Dawson (Pennsylvania), Antonio Durán (Sevilla), Solomon Feferman (Stanford), Alejandro Garciadiego (Mèxico), Ivor Grattan-Guinness (Bengeo, Herts), Jeremy Gray (Milton Keynes), Hans Niels Jahnke (Bielefield), Ignacio Janè (Barcelona), Detlef Laugwitz (Darmstadt), Herbert Mehrtens (Berlin), Volker Peckhaus (Erlangen), José F. Ruiz (Madrid), Erhard Scholz (Wuppertal). Thanks are also due to the Niedersãchsische Staats―und Universitãtsbibliothek Götingen, and particularly to Helmut Rohlfing, director of its handschriftenabteilung, for their permission to quote some unpublished material and reproduce an illustration. Another illustration comes from the Bancroft Library, University of California at Berkeley. 

𝔄𝔫𝔡 finally, how could I forget Dolores and Inés, who provided personal support and also frequent distraction from the work! 

Sevilla, July 1999 

José Ferreiròs. 

Peter Gustav Lejeune Dirichlet 1805 ― 1859

1. This danger is present, for instance, in the most comprehensive historical account of Cantor's life and work [Dauben 1979]. It is instructive to compare the partially overlapping contributions of Dauben and Hawkins to [Grattan-Guinness 1980]. 

2. The danger of excessive concentration on a single discipline is present in the excellent collection [Garatan-Guniess 1980], the multi disciplinary approach can be found. e.g., [Moore 1982], which is also the best example of collective historiography in connection to our topic. 

3. On the historiographical issue see the recent overview [Servos 1993], on the example of Berlin [Rowe 1989]. 

4. See [Rowe 2002]. An extreme view of schools as linked with single personalities was given by Hilbert in a 1922 address published by Rowe as an appendix. 

5. Another example would be the synthetic and analytic 'traditions' in geometry (formally called 'schools'). 

6. In the case of Göttingen, I could also have talked of a Göttingen tradition, since one can make the case of common methodological orientations, of an essentially abstract and modernizing kind, in a whole series of actors related to that center, from Gauss to Hilbert and Noether. 

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By Sohail Tahir